The simplest example is the xor-problem (aka either or). This was also why multilayer perceptrons as the basis of deep learning where actually created. Because a linear model cannot solve it.

Oh true, I remembered that wrong (and makes more sense this way ^^). Thx for the correction :)

Some additions / corrections:

1. Your brain works all the time. While it's using a big amount of energy for all the processing it's doing, it mainly redistributes this energy to the tasks at hand. So "thinking hard" won't increase your energy consumption that much.
2. The food we eat is not necessarily converted to glucose. While sugar are converted to glucose, fatty acis and ammino acids can either be used by neoglucogenesis (conversion to sugar) in the liver or be can be directly used as fuel by your muscles. The brain is indeed a bit special that it can only use glucose or ketone body's as fuel - ketone body as like fatty acids in disguise.

So while thinking could increase your energy uptake minimally, a more substantial influence could be the release of stress hormones (aka adrenaline) which increase the metabolic activity of the whole body